Theorem eqvrelsymb 39229

Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised and distinct variable conditions removed by Peter Mazsa, 2-Jun-2019.)

Hypothesis

Ref	Expression
eqvrelsymb.1	⊢ (𝜑 → EqvRel 𝑅)

Assertion

Ref	Expression
eqvrelsymb	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))

Proof of Theorem eqvrelsymb

Step	Hyp	Ref	Expression
1		eqvrelsymb.1	. . . 4 ⊢ (𝜑 → EqvRel 𝑅)
2	1	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → EqvRel 𝑅)
3		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4	2, 3	eqvrelsym 39228	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵𝑅𝐴)
5	1	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → EqvRel 𝑅)
6		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐵𝑅𝐴)
7	5, 6	eqvrelsym 39228	. 2 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐵)
8	4, 7	impbida 812	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   class class class wbr 5113   EqvRel weqvrel 38739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-refrel 39131  df-symrel 39163  df-trrel 39197  df-eqvrel 39208

This theorem is referenced by:  eqvrelth  39234